Orthogonality of quasi-nature spectral polynomials of Jacobi and Laguerre type

TL;DR

This paper investigates the orthogonality properties of quasi-spectral polynomials related to Jacobi and Laguerre polynomials, deriving explicit coefficients, zero locations, and interlacing properties, with extensions to unit circle polynomials.

Contribution

It provides explicit expressions for coefficients ensuring orthogonality of quasi-Christoffel and quasi-Geronimus polynomials of Jacobi and Laguerre types, and explores zero distribution and boundary support effects.

Findings

Zeros of quasi-spectral Jacobi polynomials interlace with those of Jacobi polynomials.

One zero of the quasi-polynomials lies on the boundary of the measure's support.

Explicit chain and minimal parameter sequences are derived at boundary points.

Abstract

In this work, the explicit expressions of coefficients involved in quasi Christoffel polynomials of order one and quasi-Geronimus polynomials of order one are determined for Jacobi polynomials. These coefficients are responsible for establishing the orthogonality of quasi-spectral polynomials of Jacobi polynomials. Additionally, the orthogonality of quasi-Christoffel Laguerre polynomials of order one is derived. In the process of achieving orthogonality, in both cases, one zero is located on the boundary of the support of the measure. This allows us to derive the chain sequence and minimal parameter sequence at the point lying at the end point of the support of the measure. Furthermore, the interlacing properties among the zeros of quasi-spectral orthogonal Jacobi polynomials and Jacobi polynomials are illustrated. Finally, we define the quasi-Christoffel polynomials of order one on the unit circle and analyze the location of their zeros for specific examples, as well as propose the problem in the general setup.